Applications of index theory to geometry and physics

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14544
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Shinshu University (2020-2022); Institute of Physical and Chemical Research (2019)
• Principal Investigator: Kubota Yosuke 信州大学, 学術研究院理学系, 講師 (30804075)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 非可換幾何学 / 高階指数理論 / トポロジカル相 / 作用素環論

Outline of Final Research Achievements

The main result of this research is an observation about the behavior of operators in higher index theory and some consequences obtained by applying it. The basic observation is a that operators with finite propagation on a metric space of a certain shape naturally lift to their covering space. It answers to several questions that has been asked in higher index theory, more specifically (1) index theory of codimensional 2 submanifolds, (2) obstractions to PSC metrics due to the infinite KO-bandwidth, and (3) a mathematical proof of the bulk-dislocation correspondence in 3-dimensional topological matter with a screw dislocation.

Free Research Field

非可換幾何学

Academic Significance and Societal Importance of the Research Achievements

高階指数理論は，作用素のなす空間のトポロジーを扱う抽象理論で，これまでに高度に非自明な理論的枠組を構築することに成功している．一方それに比べると，その抽象論がどのような問題に適用されうるかについての知見はまだ不足している．本研究では，分野が培ってきた理論がどのようなことを証明する能力を持っているかについて，具体的な事例の研究をもって理解を推し進めることができた．このような方向性の研究は，分野の知見を数学全体の中に根付かせるために大切である．